- #1

aheight

- 321

- 109

- TL;DR Summary
- I don't understand Wikipedia's definition of when a polynomial system has a finite or infinite solution. That is, I don't know what "pure power of a variable in the Grobner basis" means:

Wish to determine when a system of polynomials has an infinite number of solutions, that is, is not zero-dimensional. The Wikipedia article : System of polynomial equations states:

I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have a leading monomial which is a pure power of the variable.

For example, I have the system:

$$

\begin{aligned}

g(z,w)&=\left(-\frac{976 w^2}{53}+\frac{178 w z}{71}+\frac{27 w}{94}+\frac{323 z^2}{84}+\frac{271 z}{67}+\frac{199}{28}\right) \\

h(z,w)&= \left(\frac{72 w^2}{89}-\frac{502 w z}{97}-\frac{44 w}{57}+\frac{73 z^2}{5}-\frac{369 z}{73}+\frac{483}{95}\right)

\end{aligned}

$$

Which Mathematica's NSolve returns four solutions. And I can compute the Grobner basis with the code below. The Grobner basis is:

$$

\begin{aligned}

&1.15564\times 10^{38} w^4-2.74482\times 10^{37} w^3-8.43405\times 10^{37} w^2+1.37174\times 10^{37} w+1.9389\times 10^{37}\\

&4.72185\times 10^{45} w^3-7.68083\times 10^{45} w^2-1.1989\times 10^{45} w+1.74982\times 10^{45} z+2.44771\times 10^{45}

\end{aligned}

$$

I've not been able to fine a definition of "pure power of a variable of Grobner basis" and was hoping someone could explain this to me. Thanks.

The first thing to do for solving a polynomial system is to decide whether it is inconsistent, zero-dimensional or positive dimensional. This may be done by the computation of a Gröbner basis of the left-hand sides of the equations. The system isinconsistentif this Gröbner basis is reduced to 1. The system iszero-dimensionalif, for every variable there is a leading monomial of some element of the Gröbner basis which is a pure power of this variable. For this test, the best monomial order (that is the one which leads generally to the fastest computation) is usually the graded reverse lexicographic one (grevlex).

I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have a leading monomial which is a pure power of the variable.

For example, I have the system:

$$

\begin{aligned}

g(z,w)&=\left(-\frac{976 w^2}{53}+\frac{178 w z}{71}+\frac{27 w}{94}+\frac{323 z^2}{84}+\frac{271 z}{67}+\frac{199}{28}\right) \\

h(z,w)&= \left(\frac{72 w^2}{89}-\frac{502 w z}{97}-\frac{44 w}{57}+\frac{73 z^2}{5}-\frac{369 z}{73}+\frac{483}{95}\right)

\end{aligned}

$$

Which Mathematica's NSolve returns four solutions. And I can compute the Grobner basis with the code below. The Grobner basis is:

$$

\begin{aligned}

&1.15564\times 10^{38} w^4-2.74482\times 10^{37} w^3-8.43405\times 10^{37} w^2+1.37174\times 10^{37} w+1.9389\times 10^{37}\\

&4.72185\times 10^{45} w^3-7.68083\times 10^{45} w^2-1.1989\times 10^{45} w+1.74982\times 10^{45} z+2.44771\times 10^{45}

\end{aligned}

$$

I've not been able to fine a definition of "pure power of a variable of Grobner basis" and was hoping someone could explain this to me. Thanks.

Mathematica:

```
In[346]:=
g0[{z_, w_}] =
199/28 + (27 w)/94 - (976 w^2)/53 + (271 z)/67 + (178 w z)/71 + (
323 z^2)/84;
h0[{z_, w_}] =
483/95 - (44 w)/57 + (72 w^2)/89 - (369 z)/73 - (502 w z)/97 + (
73 z^2)/5;
NSolve[{g0[{z, w}] == 0, h0[{z, w}] == 0}, {z, w}]
GroebnerBasis[{g0[{z, w}], h0[{z, w}]}, {z, w}] // N
Out[348]= {{z -> 0.309562 - 0.532126 I,
w -> 0.688127 - 0.178627 I}, {z -> 0.309562 + 0.532126 I,
w -> 0.688127 + 0.178627 I}, {z -> 0.0622197 - 0.609678 I,
w -> -0.56937 + 0.088145 I}, {z -> 0.0622197 + 0.609678 I,
w -> -0.56937 - 0.088145 I}}
Out[349]= {1.9389*10^37 + 1.37174*10^37 w - 8.43405*10^37 w^2 -
2.74482*10^37 w^3 + 1.15564*10^38 w^4,
2.44771*10^45 - 1.1989*10^45 w - 7.68083*10^45 w^2 +
4.72185*10^45 w^3 + 1.74982*10^45 z}
```